where \(f_i(\mathbf {x}, t)\)  is the probability density function of fluid molecules with discrete velocity \(\mathbf {c}_i\)  at position \(\mathbf {x}\)  and at time \(t\) , \(\Delta t\)  is the discrete time interval, \(\tau \)  is the relaxation time, \(f_i^{(\mbox{\tiny eq})}\)  is the equilibrium distribution function (that is the analogous of the Maxwell distribution in the Boltzmann equation), and \(f_i^{(F)}\)  is the force density that has been implemented according to the Guo scheme [1]}. Once the probability density function \(f_i(\mathbf {x}, t)\)  has been computed, one can compute the density and the velocity fields: \(\rho (\mathbf {x}, t) = \sum _{i} f_i(\mathbf {x}, t)\)  and \(\mathbf {u}(\mathbf {x}, t) = \sum _{i} \mathbf {c}_i f_i(\mathbf {x}, t)/\rho (\mathbf {x}, t)\) , respectively [2]}.
